Chern–Weil homomorphism

In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.

Let G be a real or complex Lie group with Lie algebra ${\displaystyle {\mathfrak {g}}}$, and let ${\displaystyle \mathbb {C} [{\mathfrak {g}}]}$ denote the algebra of ${\displaystyle \mathbb {C} }$-valued polynomials on ${\displaystyle {\mathfrak {g}}}$ (exactly the same argument works if we used ${\displaystyle \mathbb {R} }$ instead of ${\displaystyle \mathbb {C} }$). Let ${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}}$ be the subalgebra of fixed points in ${\displaystyle \mathbb {C} [{\mathfrak {g}}]}$ under the adjoint action of G; that is, the subalgebra consisting of all polynomials f such that ${\displaystyle f(\operatorname {Ad} _{g}x)=f(x)}$, for all g in G and x in ${\displaystyle {\mathfrak {g}}}$,

Given a principal G-bundle P on M, there is an associated homomorphism of ${\displaystyle \mathbb {C} }$-algebras,

${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}\to H^{*}(M;\mathbb {C} )}$,

called the Chern–Weil homomorphism, where on the right cohomology is de Rham cohomology. This homomorphism is obtained by taking invariant polynomials in the curvature of any connection on the given bundle. If G is either compact or semi-simple, then the cohomology ring of the classifying space for G-bundles, ${\displaystyle BG}$, is isomorphic to the algebra ${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}}$ of invariant polynomials:

${\displaystyle H^{*}(BG;\mathbb {C} )\cong \mathbb {C} [{\mathfrak {g}}]^{G}.}$

(The cohomology ring of BG can still be given in the de Rham sense:

${\displaystyle H^{k}(BG;\mathbb {C} )=\varinjlim \operatorname {ker} (d\colon \Omega ^{k}(B_{j}G)\to \Omega ^{k+1}(B_{j}G))/\operatorname {im} d.}$

when ${\displaystyle BG=\varinjlim B_{j}G}$ and ${\displaystyle B_{j}G}$ are manifolds.)